Acute / Obtuse Triangles - SAT Math

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Question

If the average of the measures of two angles in a triangle is 75o, what is the measure of the third angle in this triangle?

Answer

The sum of the angles in a triangle is 180o: a + b + c = 180

In this case, the average of a and b is 75:

(a + b)/2 = 75, then multiply both sides by 2

(a + b) = 150, then substitute into first equation

150 + c = 180

c = 30

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Question

Which of the following can NOT be the angles of a triangle?

Answer

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

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Question

Let the measures, in degrees, of the three angles of a triangle be x, y, and z. If y = 2z, and z = 0.5x - 30, then what is the measure, in degrees, of the largest angle in the triangle?

Answer

The measures of the three angles are x, y, and z. Because the sum of the measures of the angles in any triangle must be 180 degrees, we know that x + y + z = 180. We can use this equation, along with the other two equations given, to form this system of equations:

x + y + z = 180

y = 2z

z = 0.5x - 30

If we can solve for both y and x in terms of z, then we can substitute these values into the first equation and create an equation with only one variable.

Because we are told already that y = 2z, we alreay have the value of y in terms of z.

We must solve the equation z = 0.5x - 30 for x in terms of z.

Add thirty to both sides.

z + 30 = 0.5x

Mutliply both sides by 2

2(z + 30) = 2z + 60 = x

x = 2z + 60

Now we have the values of x and y in terms of z. Let's substitute these values for x and y into the equation x + y + z = 180.

(2z + 60) + 2z + z = 180

5z + 60 = 180

5z = 120

z = 24

Because y = 2z, we know that y = 2(24) = 48. We also determined earlier that x = 2z + 60, so x = 2(24) + 60 = 108.

Thus, the measures of the three angles of the triangle are 24, 48, and 108. The question asks for the largest of these measures, which is 108.

The answer is 108.

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Question

Angles x, y, and z make up the interior angles of a scalene triangle. Angle x is three times the size of y and 1/2 the size of z. How big is angle y.

Answer

The answer is 18

We know that the sum of all the angles is 180. Using the rest of the information given we can write the other two equations:

x + y + z = 180

x = 3y

2x = z

We can solve for y and z in the second and third equations and then plug into the first to solve.

x + (1/3)x + 2x = 180

3\[x + (1/3)x + 2x = 180\]

3x + x + 6x = 540

10x = 540

x = 54

y = 18

z = 108

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Question

In the picture above, $\overline{AB}$ is a straight line segment. Find the value of .

Answer

A straight line segment has 180 degrees. Therefore, the angle that is not labelled must have:

$180-110 = 70^{\circ}$

We know that the sum of the angles in a triangle is 180 degrees. As a result, we can set up the following algebraic equation:

Subtract 70 from both sides:

Divide by 2:

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Question

Solve each problem and decide which is the best of the choices given.

Solve for .

Answer

To solve for , you must first solve for .

All triangles' angles add up to $180$^{\circ}$$ .

So subtract from to get , the value of .

Angles and are supplementary, meaning they add up to .

Subtract from to get $60$^{\circ}$$ .

$3x=60$^{\circ}$$ , so $x=20$^{\circ}$$ .

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Question

If $x^{\circ}$ , $4x^{\circ}$ , and $30^{\circ}$ are measures of three angles of a triangle, what is the value of ?

Answer

Since the sum of the angles of a triangle is $180^{\circ}$ , we know that

$x^{\circ} + 4x^{\circ} + 30^{\circ} = 180^{\circ}$ .

So

$5x^{\circ} + 30^{\circ} = 180^{\circ}$

$5x^\circ=180^\circ-30^\circ$

and $x = 30^{\circ}$ .

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Question

Figure is not drawn to scale.

Refer to the provided figure. Evaluate $m \angle CAD$ .

Answer

$\bigtriangleup ABC$ is an equilateral, so all of its angles - in particular, $\angle ACB$ - measure $60^{\circ }$ . This angle is an exterior angle to $\bigtriangleup DAC$ , and its measure is equal to the sum of those of its two remote interior angles, $\angle DAC$ and $\angle D$ , so

$m\angle DAC + m\angle D = m \angle ACB$

Setting $m \angle ACB =60^{ \circ}$ and $m \angle D = 42^{\circ }$ , solve for $m\angle DAC$ :

$m\angle DAC + 42^{\circ }= 60^{\circ }$

$m\angle DAC = 60^{\circ } - 42^{\circ }= 18^{\circ }$

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Question

Refer to the above figure. Evaluate $m \angle CDE$ .

Answer

$\bigtriangleup ABC$ is marked with three congruent sides, making it an equilateral triangle, so $m \angle ABC = 60 ^{\circ }$ . This is an exterior angle of $\bigtriangleup BCD$ , making its measure the sum of those of its remote interior angles; that is,

$m \angle BCD + m \angle BDC = m \angle ABC$

$\bigtriangleup BCD$ has congruent sides $\overline{BC}$ and $\overline{CD}$ , so, by the Isosceles Triangle Theorem, $m \angle BCD = m \angle BDC$ . Substituting $m \angle BDC$ for $m \angle BCD$ and $60 ^{\circ }$ for $m \angle ABC$ :

$m \angle BDC + m \angle BDC = 60 ^{\circ }$

$2 \cdot m \angle BDC = 60 ^{\circ }$

$m \angle BDC = 30 ^{\circ }$

$\angle BDC$ and $\angle CDE$ form a linear pair and are therefore supplementary - that is, their degree measures total $180 ^{\circ }$ . Setting up the equation

$m \angle BDC + m\angle CDE= 180 ^{\circ }$

and substituting:

$30 ^{\circ } + m\angle CDE= 180 ^{\circ }$

$m\angle CDE= 150 ^{\circ }$

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Question

$\text{Triangle A}$ and $\text{Triangle B}$ are similar triangles. The perimeter of Triangle A is 45” and the length of two of its sides are 15” and 10”. If the perimeter of Triangle B is 135” and what are lengths of two of its sides?

Answer

The perimeter is equal to the sum of the three sides. In similar triangles, each side is in proportion to its correlating side. The perimeters are also in equal proportion.

Perimeter A = 45” and perimeter B = 135”

The proportion of Perimeter A to Perimeter B is $\frac{45}{135}=\frac{1}{3}$ .

This applies to the sides of the triangle. Therefore to get the any side of Triangle B, just multiply the correlating side by 3.

15” x 3 = 45”

10” x 3 = 30“

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Question

If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?

Answer

Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.

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Question

The height, , of triangle in the figure is one-fourth the length of . In terms of h, what is the area of triangle ?

Answer

If $\dpi{100} \small h=\frac{1}{4}$ * $\dpi{100} \small \overline{PQ}$ , then the length of $\dpi{100} \small \overline{PQ}$ must be $\dpi{100} \small 4h$ .

Using the formula for the area of a triangle ( $\frac{1}{2}bh$ ), with $\dpi{100} \small b=4h$ , the area of the triangle must be $2h^{2}$ .

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Question

Find the height of a triangle if the area of the triangle = 18 and the base = 4.

Answer

The area of a triangle = (1/2)bh where b is base and h is height. 18 = (1/2)4h which gives us 36 = 4h so h =9.

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Question

Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?

Answer

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

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Question

If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?

Answer

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

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Question

A triangle has sides of length 8, 13, and L. Which of the following cannot equal L?

Answer

The sum of the lengths of two sides of a triangle cannot be less than the length of the third side. 8 + 4 = 12, which is less than 13.

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Question

Two sides of a triangle are 20 and 32. Which of the following CANNOT be the third side of this triangle.

Answer

Please remember the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Therefore, the correct answer is 10 because the sum of 10 and 20 would not be greater than the third side 32.

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Question

A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?

Answer

The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.

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Question

The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?

Answer

Let us call the third side x. According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the other two sides. Thus, all of the following must be true:

x + 7 > 9

x + 9 > 7

7 + 9 > x

We can solve these three inequalities to determine the possible values of x.

x + 7 > 9

Subtract 7 from both sides.

x > 2

Now, we can look at x + 9 > 7. Subtracting 9 from both sides, we obtain

x > –2

Finally, 7 + 9 > x, which means that 16 > x.

Therefore, x must be greater than 2, greater than –2, but also less than 16. The only number that satisfies all of these requirements is 12.

The answer is 12.

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Question

The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?

Answer

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater (not greater than or equal) than the remaining side. Thus, the following inequalities must all be true:

x + 8 > 12

x + 12 > 8

8 + 12 > x

Let's solve each inequality.

x + 8 > 12

Subtract 8 from both sides.

x > 4

Next, let's look at the inequality x + 12 > 8

x + 12 > 8

Subtract 12 from both sides.

x > –4

Lastly, 8 + 12 > x, which means that x < 20.

This means that x must be less than twenty, but greater than 4 and greater than –4. Since any number greater than 4 is also greater than –4, we can exclude the inequality x > –4.

To summarize, x must be greater than 4 and less than 20. We can write this as 4 < x < 20.

The answer is 4 < x < 20.

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